Shape-Preserving and Convergence Properties for the q-Szász-Mirakjan Operators for Fixed q∈(0,1)
We introduce a q-generalization of Szász-Mirakjan operators Sn,q and discuss their properties for fixed q∈(0,1). We show that the q-Szász-Mirakjan operators Sn,q have good shape-preserving properties. For example, Sn,q are variation-diminishing, and preserve monotonicity, convexity, and concave modu...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/563613 |
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| Summary: | We introduce a q-generalization of Szász-Mirakjan operators Sn,q and discuss their properties for fixed q∈(0,1). We show that the q-Szász-Mirakjan operators Sn,q have good shape-preserving properties. For example, Sn,q are variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixed q∈(0,1), we prove that the sequence {Sn,qf} converges to B∞,q(f) uniformly on [0,1] for each f∈C[0, 1/(1-q)], where B∞,q is the limit q-Bernstein operator. We obtain the estimates for the rate of convergence for {Sn,qf} by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. |
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| ISSN: | 1085-3375 1687-0409 |